Universal Bayes Consistency in Metric Spaces

Hanneke, Steve; Kontorovich, Aryeh; Sabato, Sivan; Weiss, Roi

doi:10.1109/ita50056.2020.9244988

Cited by 6 publications

(4 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition to restriction on the hypothesis class, there is extensive research on universal consistency under stochastic process assumptions on observations [29,1,12,11,6,5]. These works on universal learning typically utilize a k-nearest neighbor type algorithm for the constructive proof, which doesn't fully align with practical scenarios of human reasoning and scientific discover.…”

Section: Non-uniform Learningmentioning

confidence: 99%

Non-uniform Breach Evolution of Landslide Dams during Overtopping Failure

Zhou²,

Xie³

et al. 2023

Preprint

View full text Add to dashboard Cite

<p>When natural dams formed by landslides and other mass flow events are breached by the impounded water, soil materials on the surface of the dam are eroded and entrained, creating erosional features that compromise its stability. The failure of landslide dams due to breaching often results in floods and debris flow downstream. Dam breach models often assume that surface erosion is uniform (surface gradient is constant) along the breach channel, despite observations from dam breach experiments that dam surface evolution is more complex. A thorough understanding of dam surface evolution during breaching is essential for assessing flood or debris flow risk, and hence for hazard mitigation. In this study, we use experiments and numerical modelling to investigate the mechanisms that influence the non-uniform morphological evolution of landslide dam breaches. Analog landslide dam models made from poorly sorted, unconsolidated soils are subjected to various inflow discharges. It is found that although the erosion along the bed and sidewalls both affect the evolution of the breach discharge, the bed erosion is what ultimately determines how the dam profiles change during the dam breach. Breaching flow scours the breach channel resulting in erosion rates, described as a function of the difference between the flow shear stress and apparent erosion resistance, that vary at different points along the dam surface. It is found that the soil erodibility remains constant while the apparent erosion resistance increases linearly along the dam surface. The complex dam failure process is numerically modeled using depth-averaged equations which assume that the evolution of the dam profiles results from the coupled effects of erosion, entrainment, and channel bed collapse. Good agreement between the observed and modeled dam profiles demonstrates that the gradual saturation of the breach flow with entrained sediment is responsible for the linear variation of the apparent erosion resistance, which in turn contributes to the formation of the surface non-linear scouring. Our work attempts to understand the mechanisms that control the breach morphology evolution of landslide dams during breaching. A new perspective on the dam erosion process and its relationship with the sediment concentration and the apparent erosion resistance is presented here, making it possible to provide more accurate physically-based outburst flood and debris flow predictions.</p>

show abstract

Section: Non-uniform Learningmentioning

confidence: 99%

Non-uniform Breach Evolution of Landslide Dams during Overtopping Failure

Zhou²,

Xie³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The interplay between learning and compression has long been recognized, popularized by Occam's razor rule of thumb (Ariew 1976), and rigorously studied in several frameworks, including in information theory in terms of the minimum description length principal and Kolmogorov complexity (Cover 1999;Li, Vitányi et al 2008), and more recently in terms of sample compression schemes in the PAC statistical learning framework (Littlestone and Warmuth 1986;Floyd and Warmuth 1995;Graepel, Herbrich, and Shawe-Taylor 2005;Gottlieb, Kontorovich, and Nisnevitch 2014;David, Moran, and Yehudayoff 2016;Hanneke, Kontorovich, and Sadigurschi 2019;Bousquet et al 2020;Alon et al 2021).…”

Section: Introductionmentioning

confidence: 99%

On Error and Compression Rates for Prototype Rules

Kerem

Weiss

2023

AAAI

View full text Add to dashboard Cite

We study the close interplay between error and compression in the non-parametric multiclass classification setting in terms of prototype learning rules. We focus in particular on a recently proposed compression-based learning rule termed OptiNet. Beyond its computational merits, this rule has been recently shown to be universally consistent in any metric instance space that admits a universally consistent rule---the first learning algorithm known to enjoy this property. However, its error and compression rates have been left open. Here we derive such rates in the case where instances reside in Euclidean space under commonly posed smoothness and tail conditions on the data distribution. We first show that OptiNet achieves non-trivial compression rates while enjoying near minimax-optimal error rates. We then proceed to study a novel general compression scheme for further compressing prototype rules that locally adapts to the noise level without sacrificing accuracy. Applying it to OptiNet, we show that under a geometric margin condition further gain in the compression rate is achieved. Experimental results comparing the performance of the various methods are presented.

show abstract

“…Cérou and Guyader [2006] proved that, if ties are broken uniformly at random, the k m -Nearest Neighbor classifier is consistent in any Polish metric space in which the Lebesgue-Besicovitch differentiation theorem holds (see also [Forzani et al, 2012, Chaudhuri andDasgupta, 2014]). Finally, it was recently shown that is possible to combine compression techniques with 1-Nearest Neighbor classification in order achieve consistency in essentially separable metric spaces, even if the Lebesgue-Besicovitch differentiation theorem does not hold [Hanneke et al, 2019].…”

Section: Introductionmentioning

confidence: 99%

A nearest neighbor characterization of Lebesgue points in metric measure spaces

Cesari

Roberto

2021

Math. Stat. Learn.

View full text Add to dashboard Cite

The property of almost every point being a Lebesgue point has proven to be crucial for the consistency of several classification algorithms based on nearest neighbors. We characterize Lebesgue points in terms of a 1-Nearest Neighbor regression algorithm for pointwise estimation, fleshing out the role played by tie-breaking rules in the corresponding convergence problem. We then give an application of our results, proving the convergence of the risk of a large class of 1-Nearest Neighbor classification algorithms in general metric spaces where almost every point is a Lebesgue point.

show abstract

Universal Bayes Consistency in Metric Spaces

Cited by 6 publications

References 37 publications

Non-uniform Breach Evolution of Landslide Dams during Overtopping Failure

Non-uniform Breach Evolution of Landslide Dams during Overtopping Failure

On Error and Compression Rates for Prototype Rules

A nearest neighbor characterization of Lebesgue points in metric measure spaces

Contact Info

Product

Resources

About